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3 and 4 .Determinants and Matrices
normal
શ્રેણિક ${A_\lambda } = \left[ {\begin{array}{*{20}{c}}
\lambda &{\lambda - 1} \\
{\lambda - 1}&\lambda
\end{array}} \right],\lambda \in N$ હોય તો $\left| {{A_1}} \right| + \left| {{A_2}} \right| + \left| {{A_3}} \right| + ....... + \left| {{A_{300}}} \right|$ મેળવો.
A
$(299)^2$
B
$(300)^2$
C
$(150)^2$
D
$(301)^2$
Solution
$\left| {{{\rm{A}}_\lambda }} \right| = \left| {\begin{array}{*{20}{c}}
\lambda &{\lambda – 1}\\
{\lambda – 1}&\lambda
\end{array}} \right| = {\lambda ^2} – {[\lambda – 1)^2}$
$ = 2\lambda – 1$
$\therefore \left| {{{\rm{A}}_1}} \right| + \left| {{{\rm{A}}_2}} \right| + \left| {{{\rm{A}}_3}} \right| + \ldots . + \left| {{{\rm{A}}_{300}}} \right|$
$=1+3^{2}+5+\ldots . .+599 $
$= \frac{300}{2}(1+599)=(300)^{2} $
Standard 12
Mathematics
